# Metamath Proof Explorer

## Theorem pnfinf

Description: Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018)

Ref Expression
Assertion pnfinf
`|- ( A e. RR+ -> A ( <<< ` RR*s ) +oo )`

### Proof

Step Hyp Ref Expression
1 rpgt0
` |-  ( A e. RR+ -> 0 < A )`
2 nnz
` |-  ( n e. NN -> n e. ZZ )`
` |-  ( ( A e. RR+ /\ n e. NN ) -> n e. ZZ )`
4 rpxr
` |-  ( A e. RR+ -> A e. RR* )`
` |-  ( ( A e. RR+ /\ n e. NN ) -> A e. RR* )`
6 xrsmulgzz
` |-  ( ( n e. ZZ /\ A e. RR* ) -> ( n ( .g ` RR*s ) A ) = ( n *e A ) )`
7 3 5 6 syl2anc
` |-  ( ( A e. RR+ /\ n e. NN ) -> ( n ( .g ` RR*s ) A ) = ( n *e A ) )`
8 3 zred
` |-  ( ( A e. RR+ /\ n e. NN ) -> n e. RR )`
9 rpre
` |-  ( A e. RR+ -> A e. RR )`
` |-  ( ( A e. RR+ /\ n e. NN ) -> A e. RR )`
11 rexmul
` |-  ( ( n e. RR /\ A e. RR ) -> ( n *e A ) = ( n x. A ) )`
12 remulcl
` |-  ( ( n e. RR /\ A e. RR ) -> ( n x. A ) e. RR )`
13 11 12 eqeltrd
` |-  ( ( n e. RR /\ A e. RR ) -> ( n *e A ) e. RR )`
14 8 10 13 syl2anc
` |-  ( ( A e. RR+ /\ n e. NN ) -> ( n *e A ) e. RR )`
15 7 14 eqeltrd
` |-  ( ( A e. RR+ /\ n e. NN ) -> ( n ( .g ` RR*s ) A ) e. RR )`
16 ltpnf
` |-  ( ( n ( .g ` RR*s ) A ) e. RR -> ( n ( .g ` RR*s ) A ) < +oo )`
17 15 16 syl
` |-  ( ( A e. RR+ /\ n e. NN ) -> ( n ( .g ` RR*s ) A ) < +oo )`
18 17 ralrimiva
` |-  ( A e. RR+ -> A. n e. NN ( n ( .g ` RR*s ) A ) < +oo )`
19 xrsex
` |-  RR*s e. _V`
20 pnfxr
` |-  +oo e. RR*`
21 xrsbas
` |-  RR* = ( Base ` RR*s )`
22 xrs0
` |-  0 = ( 0g ` RR*s )`
23 eqid
` |-  ( .g ` RR*s ) = ( .g ` RR*s )`
24 xrslt
` |-  < = ( lt ` RR*s )`
25 21 22 23 24 isinftm
` |-  ( ( RR*s e. _V /\ A e. RR* /\ +oo e. RR* ) -> ( A ( <<< ` RR*s ) +oo <-> ( 0 < A /\ A. n e. NN ( n ( .g ` RR*s ) A ) < +oo ) ) )`
26 19 20 25 mp3an13
` |-  ( A e. RR* -> ( A ( <<< ` RR*s ) +oo <-> ( 0 < A /\ A. n e. NN ( n ( .g ` RR*s ) A ) < +oo ) ) )`
27 4 26 syl
` |-  ( A e. RR+ -> ( A ( <<< ` RR*s ) +oo <-> ( 0 < A /\ A. n e. NN ( n ( .g ` RR*s ) A ) < +oo ) ) )`
28 1 18 27 mpbir2and
` |-  ( A e. RR+ -> A ( <<< ` RR*s ) +oo )`